The perturbation scheme defined by a Rockafellian produces a relaxation of the actual problem involving minimization over both decision variables and perturbation parameters. If tuned appropriately, however, the relaxation may turn out to be equivalent to the actual problem in some sense. The tasks of tuning relaxations are traditionally referred to as dual problems, with classical linear programming and augmented Lagrangian duality emerging from specific choices of Rockafellians; see [1]} and [2]}. We focus on perturbations that stem from ambiguity about probability distributions in stochastic optimization. This brings us beyond classical perturbation schemes, with tuning becoming less critical. In fact, our development is related to penalty methods and especially exact penalization in constrained optimization (see for example [3]}, [4]}). This is reflected in our choice of terminology: exact and asymptotically exact Rockafellians emerge here for the first time.
